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On the quantum differential equations for a family of non-Kähler monotone symplectic manifolds

Kai Hugtenburg

Abstract

In this paper we prove Gamma Conjecture $1$ for twistor bundles of hyperbolic $6$ manifolds, which are monotone symplectic manifolds which admit no Kähler structure. The proof involves a direct computation of the $J$-function, and a version of Laplace's method for estimating power series (as opposed to integrals). This method allows us to rephrase Gamma Conjecture $1$ in certain situations to an Apéry-like discrete limit. We use this to give a simple proof of Gamma Conjecture $1$ for projective spaces. Additionally we show that the quantum connections of the twistor bundles we consider have unramified exponential type.

On the quantum differential equations for a family of non-Kähler monotone symplectic manifolds

Abstract

In this paper we prove Gamma Conjecture for twistor bundles of hyperbolic manifolds, which are monotone symplectic manifolds which admit no Kähler structure. The proof involves a direct computation of the -function, and a version of Laplace's method for estimating power series (as opposed to integrals). This method allows us to rephrase Gamma Conjecture in certain situations to an Apéry-like discrete limit. We use this to give a simple proof of Gamma Conjecture for projective spaces. Additionally we show that the quantum connections of the twistor bundles we consider have unramified exponential type.
Paper Structure (12 sections, 35 theorems, 123 equations)

This paper contains 12 sections, 35 theorems, 123 equations.

Table of Contents

  1. Introduction
  2. Rephrasing Gamma Conjecture $1$
  3. Acknowledgements
  4. The Gamma class and the loop space
  5. Quantum differential equations
  6. Laplace's method for power series
  7. Gamma Conjecture $1$ for $\mathbb{CP}^N$
  8. The $J$-function for twistor bundles
  9. Proof of Gamma Conjecture $1$ for the twistor bundles
  10. Unramified exponential type
  11. The irregularity number
  12. The quantum $\mathcal{D}$-module of twistor bundles

Key Result

Theorem A

Let $Z$ be the twistor bundle associated to a hyperbolic $6$ manifold with vanishing Stiefel-Whitney classes. Then $Z$ satisfies Gamma Conjecture $1$.

Theorems & Definitions (73)

  • Definition 1.1
  • Definition 1.2
  • Remark 1.3
  • Theorem A
  • Theorem B
  • Remark 1.4
  • Remark 1.5
  • Lemma 1.6
  • Definition 1.7
  • Definition 1.8
  • ...and 63 more